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## Spiraler, Fibonacci og det at være en plante

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# Kruseduller i matematiktimen: Spiraler, Fibonacci, og at være en plante [2 af 3]

## Video udskrift

Say you're me and
you're in math class, and you're doodling
flowery petally things. If you want something with
lots of overlapping petals, you're probably following
a loose sort of rule that goes something like this. Add new petals where there's
gaps between old petals. You can try doing
this precisely. Start with some number
of petals, say five, then add another
layer in between. But the next layer, you have to
add 10, then the next has 20. The inconvenient part
of this is that you have to finish a layer
before everything is even. Ideally, you'd have
a rule that just lets you add petals
until you get bored. Now imagine you're
a plant, and you want to grow in a
way that spreads out your leaves to catch the
most possible sunlight. Unfortunately,
and I hope I'm not presuming too much in
thinking that, as a plant, you're not very smart. You don't know how to add
number to create a series, you don't know geometry
and proportions, and can't draw spirals, or
rectangles, or slug cats. But maybe you could
follow one simple rule. Botanists have
noticed that plants seem to be fairly
consistent when it comes to the angle between
one leaf and the next. So let's see what you
could do with that. So you grow your first leaf,
and if you didn't change angle at all, then the
next leaf you grow would be directly above it. So that's no good,
because it blocks all the light or something. You can go 180 degrees, to
have the next leaf directly opposite, which seems ideal. Only once you go 180
again, the third leaf is right over the first. In fact, any
fraction of a circle with a whole number
as a base is going to have complete overlap
after that number of turns. And unlike when you're
doodling, as a plant you're not smart enough to see
you've gone all the way around and now should switch to
adding things in between. If you try and
postpone the overlap by making the
fraction really small, you just get a ton of
overlap in the beginning, and waste all this space,
which is completely disastrous. Or maybe other
fractions are good. The kind that position leaves
in a star like pattern. It will be a while
before it overlaps, and the leaves will be more
evenly spaced in the meantime. But what if there
were a fraction that never completely overlapped? For any rational fraction,
eventually the star will close. But what if you used
an irrational number? The kind of number that can't
be expressed as a whole number ratio. What if you used the
most irrational number? If you think it sounds weird
to say one irrational number is more irrational than
another, well, you might want to become
a number theorist. If you are a number
theorist, you might tell us that phi is
the most irrational number. Or you might say, that's like
saying, of all the integers, 1 is the integer-iest. Or you might
disagree completely. But anyway, phi. It's more than 1, but less
than 2, more than 3/2, less than 5/3. Greater than 8/5,
but 13/8 is too big. 21/13 is just a
little too small, and 34/21 is even closer,
but too big, and so on. Each pair of adjacent
Fibonacci numbers creates a ratio that
gets closer and closer to Phi as the numbers increase. Those are the same numbers on
the sides of these squares. Now stop being a
number theorist, and start being a plant again. You put your first leaf
somewhere, and the second leaf at an angle, which is
one Phi-th of a circle. Which depending
on whether you're going one way or the other,
could be about 222.5 degrees, or about 137.5. Great, your second leaf is
pretty far from the first, gets lots of space in the sun. And now let's add the next one
a Phi-th of the circle away. And again, and again. You can see how new
leaves tend to pop up in the spaces left
between old leaves, but it never quite
fills things evenly. So there's always room
for one more leaf, without having to do
a whole new layer. It's very practical and as a
plant you probably like this. It would also be a good
way to give lots of room to seed pods and
petals and stuff. As a plant that
follows this scheme, you'd be at an advantage. Where do spirals come in? Let's doodle a pinecone
using this same method. By the way, you can make
your own phi angle-a-tron by dog-earing a corner
of your notebook. If you folded it so
the edges of the line-- You have 45 degrees
plus 90, which is 135. Pretty close to 137.5. If you're careful, you can
slip in a couple more degrees. Detach your angle-a-tron
and you're good to go. Add each new pine cone-y
thing a phi angle around, and make them a little
farther out each time. Which you can keep
track of by marking the distance on
your angle-a-tron. Check it out! The spirals form by themselves. And if we count the number of
arms, look it's five and eight. If you're wondering
why spirals would form, and why always with
Fibonacci numbers, you could morph back into a
number theorist, or a geometer or something. But here's just a
little bit of intuition. One simple way to do
a flower is to start with a certain number
of petals, say five. And when you go back
around, add the next layer close to the first, but bigger. Each layer adds five new petals
and the five arms spiral out. Looks pretty spiral-ey to me. Now go back to phi. You put out three petals
before you go back around. And if I make them
really wide, the next lap adds three petals that overlap
a bunch with the first, and so on. If I started with
skinnier petals though, the second time you go around it
doesn't quite overlap so much. And it takes eight petals
before it goes around twice and they overlap enough
for you to see the spirals. So this time I get 8 and 13. I mean none of the spirals
are actually physically there on any of these plants,
it's just the plant bits are close enough that
you can see the pattern. So all the plant needs to
do to get awesome Fibonacci numbers of spirals, is add new
bits at 137.5 degree angles. The rest takes care of itself. That the Fibonacci series
is in so many things really says less
about those things, and more about mathematics. I mean that's what
mathematics is all about. Simple rules,
complex consequences. A process so easy that
even a plant can do it, can turn into these amazing
structures all around us. Just like a few
simple postulates can give us an incredibly
powerful geometry. I mean that's all assuming
that a plant can do it. But measuring the angles
between plant bits, you can see they obviously
do do it, somehow. I mean, it's not like
they have angle-trons, but plants have been
around a long time, and have had a lot of
practice, so that probably explains everything. And so we always get spirals of
5 and 8 on this flower, 5 and 8 on this artichoke, 5
and 8 on this pine cone, even this cauliflower has 1,
2, 3, 4, 5, 6, 7-- um, anyway, we always get 1, 2
3, 4, 5, 6, 7-- Huh. And 1, 2, 3, 4? It's easy to dismiss
these as mutant anomalies, but just because they're
different and unusual, doesn't mean we
should ignore them. 4? 7? 11? What could these numbers mean? Maybe things aren't as
simple as I thought.